To put it in terms first suggested by the now-Chief Justice, the constitution is "not some holy grail which only judicial initiates of the superior courts may touch" (see para. 70).
Once a body is vested with the express or implied power to determine questions of law, it is empowered to determine constitutional questions (subject to de novo review by the regular courts), barring express legislative exclusion of the authority to consider constitutional questions. Moreover, in its 2010 decision in Conway, the Supreme Court of Canada extended this logic to remedies (see para. 80), although with the additional question "whether the tribunal can grant the particular remedy sought, given the relevant statutory scheme" (see para. 82).
An excellent recent example of these principles in action is provided by Noël J.'s judgment in Canada (Procureur général) c. Telbani, 2012 CF 474. Telbani had complained to the Canadian Security Intelligence Service (CSIS) about their agents' warrantless search of his premises and aggressive treatment of him in violation of his Charter rights. This alleged behaviour lead up to Telbani being placed on a no-fly list. Finding no joy with CSIS, Telbani proceeded to make a complaint to the Security Intelligence Review Committee (SIRC), asking them to conduct an investigation. Cleverly (for reasons which will become apparent later), he also requested SIRC to recommend to CSIS that all negative reports about him be scrubbed, his name be removed from the no-fly list, and damages paid. SIRC concluded that it had jurisdiction to grant the remedies sought.
CSIS applied for judicial review of this determination. The core of its argument was that SIRC only had the power to issue recommendations. None of its findings were binding on CSIS. To draw an analogy, sometimes it is argued that procedural fairness does not apply in respect of decisions that are preliminary or merely administrative in nature. CSIS was, in essence, relying on this logic, which underpinned the conclusions of Cory J. in Thomson v. Canada (Deputy Minister of Agriculture),  1 S.C.R. 385:
The Committee's recommendation constitutes a report put forward as something worthy of acceptance. It serves to ensure the accuracy of the information on which the Deputy Minister makes the decision, and it gives the Deputy Minister a second opinion to consider. It is no more than that.Such purely recommendatory functions could not, it was argued, ground a general power to answer constitutional questions or grant constitutional remedies. In the absence of an express power to answer questions of law, none could be implied.
Noël J. was not impressed by this submission. On reviewing SIRC's statutory mandate, he concluded that its very purpose was to ensure that CSIS remained within the bounds of legality:
Given this mission, SIRC simply had to have an implicit power of legal interpretation: La préoccupation constante de tous ces contrôles est, dans la mesure du possible, de s’assurer que le SCRS opère de façon continue selon les lois du Canada et leurs règlements et qu’il n’exerce pas ses pouvoirs de façon abusive ou inutile. Les pouvoirs du SCRS sont considérables, mais malgré ce large pouvoir qu’il a attribué, le législateur a voulu s’assurer que les droits fondamentaux soient protégés. Lorsqu’on a à apprécier le rôle d’enquête du CSARS, il est donc important d’avoir à l’esprit ce souci du législateur de s’assurer que le mandat du SCRS est articulé de façon légale et qu’il ne va pas à l’encontre des lois et règlements applicables en matières semblables.
This conclusion was bolstered by reference to the court-like powers that SIRC possesses (see para. 123). Finally, because the remedies sought were recommendatory in nature, there was no statutory barrier to SIRC issuing them. Declaring that CSIS had acted in contravention of Mr. Telbani's Charter rights would fall comfortably within SIRC's mandate. A more hard-edged remedy might not have. Il est donc évident que pour accomplir ses fonctions d’enquête, le CSARS ne peut tout simplement opérer dans un vacuum juridique. Il se doit d’interpréter le droit pour trancher les questions de fait qui lui sont soumises par l’entremise des dossiers de plaintes. Dire l’inverse irait à l’encontre du souci du législateur de s’assurer que le SCRS opère de façon conforme aux lois du Canada, dont la LSCRS, ses règlements et les politiques du SCRS.
The takeaway from Telbani is that an implicit power to answer questions of law will almost always be present. Even where a body's function is purely recommendatory, such recommendations cannot be made, as Noël J. put it, in a "juridical vacuum". Indeed, even where the powers in issue are purely administrative or preliminary in nature, the decision-maker will presumably need to take some view of the relevant legal provisions. Trying to fight cases on the basis that there is no implicit power to answer questions of law seems like a losing proposition.
This suggests that, in determining the jurisdiction of administrative bodies to grant constitutional remedies, the real action will be in (1) determining if there is an express statutory exclusion of the power to consider constitutional questions and (2) determining whether the remedy sought would contradict the body's statutory mandate. Given that the presence of (1) will be immediately apparent to all, maybe the only action will ultimately be at (2). Counsel should be as clever as Mr. Telbani's in carefully crafting their requests for constitutional remedies from administrative bodies.